Question

1.将(2+e^x)^2展开成x的幂级数 2.将函数f(x)=1/(x^2+3x+2)展开成关于（x-1）的幂级数

Answer 1

1)

(2+e^x)^2=4+4e^x+e^(2x)

=4+4(1+x+x^2/2!+..x^n/n!+..)+(1+2x+2^2x^2/2!+..+2^n*x^n/n!+..)

=9+6x+4x^2+...+(4+2^n)x^n/n!+...

2)

f(x)=1/[(x+1)(x+2)]=1/(x+1)-1/(x+2)=1/(x-1+2)-1/(x-1+3)

=1/2*1/[1+(x-1)/2]-1/3*1/[1+(x-1)/3]

=1/2*[ 1-(x-1)/2+(x-1)^2/2^2+...+(-1)^n(x-1)^n/2^n+...]-1/3*[1-(x-1)/3+(x-1)^2/3^2-.+(-1)^n(x-1)^n/3^n+.]

=1/6-(1/4-1/9)*(x-1)+.+ (-1)^n[1/2^(n+1)-1/3^(n+1)](x-1)^n+.